Practical challenges in data‐driven interpolation: Dealing with noise, enforcing stability, and computing realizations

Abstract

SummaryIn this contribution, we propose a detailed study of interpolation‐based data‐driven methods that are of relevance in the model reduction and also in the systems and control communities. The data are given by samples of the transfer function of the underlying (unknown) model, that is, we analyze frequency‐response data. We also propose novel approaches that combine some of the main attributes of the established methods, for addressing particular issues. This includes placing poles and hence, enforcing stability of reduced‐order models, robustness to noisy or perturbed data, and switching from different rational function representations. We mention here the classical state‐space format and also various barycentric representations of the fitted rational interpolants. We show that the newly‐developed approaches yield, in some cases, superior numerical results, when comparing to the established methods. The numerical results include a thorough analysis of various aspects related to approximation errors, choice of interpolation points, or placing dominant poles, which are tested on some benchmark models and data‐sets.

Plug‐and‐play adaptive surrogate modeling of parametric nonlinear dynamics in frequency domain

“…The number $$$ {S}_j $$$ and the locations of the support frequencies $$$ {\omega}_{j,i} $$$ are crucial for the approximation properties and efficiency of the MRI surrogate: too few samples yield an inaccurate approximation, whereas too many make the training of the local surrogate overly expensive, and can even result in numerical instabilities 20,35 . A practical and effective way of choosing the support frequencies is the greedy MRI (gMRI) method, introduced in Reference 19.…”

Structured barycentric forms for interpolation-based data-driven reduced modeling of second-order systems

“…In the non-intrusive case, for methods that are based on barycentric forms and matching transfer function values, there have been some attempts to impose stability as a post-processing tool such as in [15,22] for the Loewner framework or in [17] for the AAA algorithm. Moreover, for the same method, the placing of stable poles was explored in [4], while optimization approaches were applied in [14].…”

“…Typically, it is not possible to extract additional differential structures from general rational functions. For example, even though one can always convert the structured transfer function in (2) to an unstructured rational function in (4), the reverse direction is not guaranteed [18,50]. Most methods for learning transfer functions from frequency domain data have been mainly developed for the unstructured case (3).…”

Leveraging the Hankel norm approximation and data‐driven algorithms in reduced order modeling

“…Though, to make it work, one has to find a system realization $$$ \left(\overset{\widetilde }{A},\tilde{B},\tilde{C},\tilde{D}\right) $$$ of the rational approximation $$$ {R}_d $$$. The block‐AAA algorithm is a good candidate for the data‐driven part because it is empirically stable and accurate and a system realization formula of its output function was derived in Reference 29. In this section, we discuss the numerical issues of using the block‐AAA algorithm for the data‐driven stage and how to rectify them by introducing a regularizer.…”

“…For the output of the AAA algorithm to be useful for the HNA algorithm, we must find a system realization of the rational approximation. The system generated by the canonical block‐AAA algorithm 26,29 has a well‐conditioned transfer function $$$ \tilde{G} $$$ but a system realization $$$ \left(\overset{\widetilde }{A},\tilde{B},\tilde{C},\tilde{D}\right) $$$ that is hard to represent in floating‐point arithmetic, causing numerical issues in the second stage (see Subsection 5.2). We propose a modification of the block‐AAA algorithm that sacrifices some accuracy for numerical stability.…”